All Questions
Tagged with abelian-groups gr.group-theory
141 questions
32
votes
3
answers
3k
views
Is there a nice explanation for this curious fact about cyclic subgroups?
Here's something that I noticed that quite surprised me.
Let $G$ be a finite abelian group. Consider the following expression.
$$
\nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H|
$$
It ...
- 6,290
2
votes
1
answer
105
views
Cotorsion-freeness in uncountable products of abelian groups
An abelian group $A$ is cotorsion provided that whenever $A \leq G$ with $G$ abelian and $G/A$ is
torsion-free, we have $G \cong A \oplus B$ for some $B \leq G$. An abelian group $A$ is
cotorsion-...
- 517
0
votes
1
answer
207
views
Fourier transform on lattice strip
I am working with a triangular lattice $L=\{n_1 a_2 + n_2 a_2 : n\in\mathbb{Z}^2 \}$ and $a_1 = \pmatrix{1 \\ 0}$ and $a_2 = \frac{1}{2} \pmatrix{-1 \\ \sqrt{3}}$, and I want to compute the Pontryagin ...
- 595
4
votes
1
answer
498
views
Short exact sequence $0\to \mathbb Z\to A \to \mathbb R \to 0$
Does every short exact sequence $0\to \mathbb Z\to A \to \mathbb R \to 0$ split in the category of Abelian groups?
- 51
6
votes
1
answer
337
views
Can all proper sublattices of $\mathbb{Z}^n$ be generated cyclically?
Let $\Lambda \subset \mathbb{Z}^n$ be a proper sublattice (so that $\Lambda \ne \mathbb{Z}^n$). We say that $\Lambda$ is cyclically generated if there exists a matrix $M \in \text{GL}_n(\mathbb{Z})$ ...
- 27k
4
votes
1
answer
405
views
A question on bi-character of finite abelian group
Setting: $G$ is a finite abelian group and any bicharacter on $G$, where a bi-character on $G$ is a map $b:G \times G \to \mathbb{Q}/\mathbb{Z}$ such that $$b(x+y,z)=b(x,z)+b(y,z),b(x,z+y)=b(x,z)+b(x,...
- 41
12
votes
2
answers
2k
views
Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?
Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?
If not, is there an example?
- 307
1
vote
1
answer
344
views
Bound for order of a group depending on number of elements of maximal order
This question has been partly answered in MSE, see here.
In a paper On the Number of Elements of maximal order in a Group, it is proven that an arbitrary group $G$ with a finite number of elements of ...
- 181
-2
votes
1
answer
158
views
Are isomorphic quotients of abelian groups induced by automorphisms? [closed]
If I have an (abelian) group $G$ and an automorphism $\sigma: G \to G$ then for any subgroup $H$ of $G$ there is an induced isomorphism $G/H \cong G/\sigma(H)$ given by the map $gH \mapsto \sigma(g)\...
- 13
0
votes
1
answer
557
views
Abelian groups and their subgroups
It is well known that every finite abelian group is a direct product of cyclic groups. So for every $n$ every finite abelian group of exponent $n$ is a direct product of cyclic groups of order at most ...
user158834
15
votes
1
answer
567
views
Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$
Is there a torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z \oplus \Bbb Z$ (as groups)?
Notice that $\Bbb Z$ is not cancellable, so
$A \oplus \Bbb Z \...
- 1,742
20
votes
4
answers
2k
views
Categorical proof subgroups of free groups are free?
This is a crossport of this question from MSE.
Is there a categorical proof that subgroups of free groups are free?
How about the result that subgroups of free abelian groups are free abelian?
What ...
- 935
13
votes
1
answer
791
views
How nearly abelian are nilpotent groups?
It is not uncommon to read that "nilpotent groups are 'close to abelian'."1,2
Can this sentiment be made precise
in the sense of the
Turán and Erdős definition of "the probability that two elements of ...
- 151k
5
votes
2
answers
907
views
What is $\mathrm{Hom}(\mathbb{Q},\mathbb{Z}(p^{\infty}))$?
What is $\mathrm{Hom}(\mathbb{Q},\mathbb{Z}(p^{\infty}))$?
I have a reference that says the group in question is $\mathbb{Q}_p,$ the additive group of the quotient field of the $p$-adic integers. Can ...
- 549
1
vote
1
answer
276
views
How many elements of each order are there in this $p$-group? [closed]
Let $G$ be a countable Abelian $p$-group which equals a direct sum of at most countably many finite cyclic groups and at most countably many copies of the Prüfer $p$-group, where these finite cyclic ...
- 4,597
6
votes
1
answer
417
views
Finite-by-torsion-free abelian groups (or compact abelian groups with finitely many components)
Here's a question I should know the answer to but don't:
Suppose $1\to F \to G \to G/F \to 1$ is a short exact sequence of abelian groups with $F$ finite and $G/F$ torsion-free. Must the sequence ...
- 9,730
3
votes
0
answers
138
views
The group of sequences in $G^{\mathbb{N}}$ that converge to $(0,0,0,\dots)$
Let $G$ be a discrete abelian group and $G^{\mathbb{N}}$ be the direct product (with the product topology), which consists of sequences $(a_1,a_2,a_3,\dots)$ of elements of $G$.
Let $G^{\infty}$ ...
- 517
10
votes
1
answer
2k
views
A group whose automorphism group is cyclic
Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
This question was first posted here.
- 599
26
votes
3
answers
7k
views
Subgroups of a finite abelian group
Let $$G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n}$$ be any finite abelian group.
What are $G$'s subgroups? I can get many subgroups by grouping the factors and multiplying them by ...
uuu
9
votes
2
answers
1k
views
On the existence of a direct summand containing a fixed subgroup
Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups that they generate are in direct sum $\langle g_1 \rangle \oplus \cdots \oplus \langle g_n \rangle$. Is it ...
- 133
6
votes
4
answers
2k
views
Is a torsion-free abelian group finitely generated, if all of its localizations at primes $p$ are finitely generated over $\mathbb{Z}_p$?
Background: When proving that the group of $k$-isogenies $\mathrm{Hom}_k(A,B)$ between two abelian varieties is finitely generated, one first shows that the Tate map $$\mathbb{Z}_\ell\otimes_{\mathbb{...
- 1,527
1
vote
1
answer
133
views
Irreducible non-Abelian subgroup of $\mathrm{U}_n(\mathbb{C})$, containing diagonal matrices
Consider an irreducible non-Abelian subgroup $\mathrm{H}$ of group of unitary matrices $\mathrm{U}_n(\mathbb{C})$, that contains the subgroup of diagonal matrices. Does there exist any result ...
- 85
8
votes
0
answers
309
views
A $\mathsf{ZF}$ example of a nonreflexive group which is isomorphic to its double dual?
Given a group $G$ denote by $G^\ast=\mathrm{Hom}(G,\Bbb Z)$ its dual and by $j\colon G\to G^{\ast\ast}$ the canonical homomorphism $g\mapsto (f\mapsto f(g))$. A group is reflexive iff $j$ is an ...
- 3,011
1
vote
1
answer
329
views
Is there a free profinite abelian group on a profinite set?
Let $\mathit{Profinite}_{\mathrm{Ab}}$ be the category of profinite abelian groups, and let $\mathit{Profinite}_{\mathrm{Set}}$ be the category of profinite sets. Does the forgetful functor
$$\mathit{...
2
votes
2
answers
4k
views
Any factor group of a finite abelian group is isomorphic to some subgroup
If you visit this link, you'll see at the top of the PDF view. Basic properties of finite abelian groups:
Every quotient group of a finite abelian group is isomorphic to a subgroup.
If the above ...
5
votes
2
answers
387
views
Size of distinct sums in A
Let $G$ be an abelian group. Let $A\subset G$ be a finite set. $\sum_A$ is defined as: $$\left\{\sum_{b\in B}b \mid B\subset A\right\}$$ Is there any result similar to Freiman's Theorem for $\sum_A$? ...
- 901
3
votes
1
answer
293
views
Abelian torsion-free group with $\mathbb{Z}_2\times\mathbb{Z}$ as automorphism group
Let $A$ be an abelian torsion-free group. If $A$ is isomorphic with the group of rational numbers whose denominators are powers of, say, $2$, then its automorphism group is isomorphic with $\mathbb{Z}...
- 287
4
votes
2
answers
507
views
Co-finite type abelian groups
Suppose $B$ is an abelian group such that for every integer $n\ge 1$, the $n$-torsion subgroup $B[n]$ is finite.
Let $B_{\rm tor} = \varinjlim_{n\ge 1} B[n]$ be the torsion subgroup of $B$.
Is it ...
user119470
8
votes
2
answers
501
views
Exact sequence of $n$th powers of abelian groups
Let $A,B,C$ be finitely generated abelian groups. Assume that there is an exact sequence $$0 \to C \to A^n \to B^n \to 0,$$where $A^n = A \oplus \dotsc \oplus A$ as usual. It is not assumed that $A^n \...
- 63.1k
4
votes
2
answers
449
views
Number of torsion-free abelian groups
Let $\mathfrak{c}$ be the cardinality of the continuum. How much Choice, if any, is needed to prove that there are $2^{\mathfrak{c}}$ distinct (mutually nonisomorphic) torsion-free abelian groups of ...
- 2,049
3
votes
1
answer
153
views
On decomposition of finite Abelian groups
It is easy to see that for any finite Abelian group $G$ and any numbers $a,b$ with $|G|=ab$ there exist a subgroup $A\subset G$ and a subset $B\subset G$ such that $|A|=a$, $|B|=b$ and $G=A+B$, where $...
- 42k
14
votes
1
answer
696
views
$\mathbb{Z}$-module structure of the subring generated by an algebraic number
Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\...
- 151
2
votes
0
answers
70
views
Alternating $n$-homomorphism on abelian group is skew of $n$-cocycle
Let $A$ be a finitely generated abelian group. Let $c$ be a 2-cocycle on $A$, where $A$ acts trivially on $\mathbb{C}^\times$. It is well-known that the skew-map
$$ c(a_1,a_2) \longmapsto \frac{c(a_1,...
- 1,813
3
votes
0
answers
98
views
Hales' generalization of the stacked bases theorem (seeking a proof)
In his paper Analogues of the stacked bases theorem, published in the proceedings of a 1976 conference, A.W. Hales claimed some interesting generalizations of the stacked bases theorem for abelian ...
- 2,992
4
votes
1
answer
383
views
Finite dimensional compact abelian group that is not a product of connected and a totally disconnected
Let $G$ be a compact abelian group. A compact abelian group is said to have dimension $n$ if $\dim_\mathbb{Q} \mathbb{Q}\otimes \hat G = n$. Equivalently one can show that this holds if $G$ is ...
- 43
5
votes
1
answer
163
views
Characteristically simple locally compact abelian groups
Say a topological group $G$ is topologically characteristically simple if there does not exist a closed subgroup $1 < K < G$ such that $K$ is invariant under all automorphisms of $G$ (here `...
- 4,738
2
votes
1
answer
190
views
Logic article on first-order invariants of abelian groups
I remember reading an article published in the 1970s by a Polish mathematician describing the first-order invariants of a torsion-free abelian group. I do not recollect the author's name, the title of ...
- 116
9
votes
1
answer
3k
views
Automorphism group of a finite group
I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = \mathbb{Z}/m\...
- 675
2
votes
2
answers
474
views
Non-archimedean group over the reals
I have a totally ordered group $(\mathbb{R};\leq,\oplus,0,-)$ with the reals as base set satisfying monotonicity, i.e.
for all $x,y,z$ we have that if $y\leq z$ then $x\oplus y \leq x\oplus z$, and I ...
- 43
3
votes
1
answer
125
views
Uncountable divisible groups and the existence of order-preserving isomorphisms of their subsets
Let $(G,+,0,<)$ be an ordered divisible group of uncountable dimension. Consider the subset $G^{<0}$ of $G$.
Question: Are $G$ and $G^{<0}$ isomorphic as ordered sets? Does there exist an ...
- 33
3
votes
1
answer
172
views
Compatible total orderings of the group $\mathbb{Z}^\mathbb{N}$
Given the additive group of the module $\mathbb{Z}^\mathbb{N}$ and a total ordering of the group that is compatible with addition and where $\chi_{\{n\}} > 0$ for all $n \in \mathbb{N}$, can we say ...
1
vote
0
answers
74
views
Pure (ordered) subgroups
Let $H,G$ be abelian groups with $H \leq G$. We say that $H$ is a pure subgroup of $G$ if for every $n \in \mathbb N$ and $h \in H$ the following holds: If $h$ is $n$-divisible in $G$, then $h$ is $n$-...
- 111
8
votes
0
answers
1k
views
Examples of uncountable abelian $p$-groups
Does anyone know of any interesting examples of an infinite abelian $p$-group which is uncountable?
By non-interesting here I mean the direct sums of cylic and quasi-cylic groups, and totally ...
7
votes
1
answer
266
views
Positive cone of a subgroup of $\mathbb{Z}^n$
This question sounds like it should be very well known, but for some reason I failed to find a decent answer anywhere. Let $G\subset\mathbb{Z}^n$ be a subgroup, and $G_+=G\cap\mathbb{Z}_{\ge0}^n$ be a ...
- 16.9k
17
votes
1
answer
1k
views
Epimorphisms $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ are split
Consider the additive group of integer sequences $\mathbb{Z}^{\mathbb{N}}$. Why does every epimorphism of groups $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ split? $(\star)$
Actually this ...
- 63.1k
8
votes
1
answer
3k
views
On order of subgroups in abelian groups
I wonder whether any of you guys has already read the homonymous note by R. Beals in the December 2009 issue of the Monthly.
If so, would you be so kind as to let me know about the main ideas in Beal'...
- 8,842
7
votes
1
answer
617
views
Looking for a modern source about Ulm Invariants
I'm looking for a modern, approachable text (preferably a website, textbook, or expository article, and preferably one easily available online or at a library) which can explain the concept of Ulm ...
- 1,979
29
votes
0
answers
877
views
The field of fractions of the rational group algebra of a torsion free abelian group
Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions.
...
- 35.2k
9
votes
3
answers
3k
views
Why are divisible abelian groups important?
I just quote wikipedia:
"Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups."
I am asking for detail ...
3
votes
2
answers
318
views
Character kernels in the lattice of subgroups of a finite abelian group
I am looking for any efforts that have been made to characterize the character kernels (equivalently, the subgroups yielding cyclic quotients) inside the lattice of subgroups of a finite abelian group....
- 2,851